Keh-Jim Dunn

Low-field NMR determinations of the properties of heavy oils and water-in-oil emulsions

Low-field (< 50 mT) nuclear magnetic resonance (NMR) well-logging measurements are beginning to be used to obtain estimates of oil viscosity in situ. To build an interpretive capability, we made laboratory T1 and T2 relaxation measurements on a suite of high-density, high-viscosity crude oils. These measurements were also used to estimate oil viscosity and water fraction from T1 and T2 measurements on stable, water-in-oil emulsions. High-density, high-viscosity oils have components that relax faster than can be measured by nuclear magnetic resonance logging tools. This requires corrections to T2 logging measurements for accurate estimates of oil saturation and porosity.

Permeability relation with other petrophysical parameters for periodic porous media

We modeled permeability (k) estimation based on porosity (ϕ), electrical formation factor (F), and nuclear magnetic resonance (NMR) relaxation time (T), using periodic structures of touching and overlapping spheres. The formation factors for these systems were calculated using the theory of bounds of bulk effective conductivity for a two‐component composite. The model allowed variations in grain consolidation (degree of overlap), scaling (grain size), and NMR surface relaxivity. The correlation of the permeability (k) with the predictor a aTbFc was slightly higher than aTbϕc (i.e., a correlation coefficient of 0.98 versus 0.95). The exponent b ranged from 1.4 for a pure grain consolidation system to 2 for a pure scaling system. Variations in surface relaxivity are shown to cause significant scatter in the correlations.

Self diffusion of nuclear spins in a porous medium with a periodic microstructure

A Fourier approach is developed for evaluating the diffusion eigenstates and the diffusion propagator of a periodic, fluid filled porous medium, and is applied to the calculation of the pulsed‐field‐gradient‐spin‐echo amplitude M(k,t). The method is most effective for long times t, but works quite well down to times that are short enough so that asymptotic short time approximations of the diffusion process are still valid. The main advantage of the method is that it is applicable regardless of the value of the porosity or the shape of the periodic pore space. It is used to calculate M(k,t) for a number of examples of periodic porous media with porosities as low as 10%.

PERMEABILITY RELATION FOR PERIODIC STRUCTURES

The permeability relation for periodic porous media is studied with respect to other petrophysical parameters such as formation factor, porosity, surface-to-volume ratio, and nuclear magnetic resonance (NMR) relaxation time. All these quantities were computed for periodic structures of simple, body-centered, and face-centered cubic arrays of touching and overlapping spheres. The formation factors were calculated by using a method which is based on a Fourier-space representation of an integral equation for the electric potential in a two-component composite. The nuclear magnetic resonance relaxation time for the case where surface-enchanced relaxation plays a dominant role is known to be V P/rho S (VP is the pore volume, S is the pore surface, is the surface relaxation strength) when rho is not too large. Previously calculated permeabilities for these structures from the literature were used for correlation studies with other petrophysical parameters. Various correlation schemes among these quantities, such as k = aTbFc, and k = aTb phi c, were investigated, where k is permeability, T is the NMR relaxation time, phi is the porosity, and F is the formation factor.

Magnetic susceptibility contrast induced field gradients in porous media

Using a two-component composite theory, we compute the internal field gradient of a periodic porous medium induced by the magnetic susceptibility contrasts. The magnetization of such a system is computed by using the diffusion eigenstates in Fourier representation. We show that the volume averaged field gradient, when used in the formula for free diffusion, significantly overestimates the magnetization decay rate. We also establish bounds for such a periodic system within which the Gaussian approximation is valid for diffusion of spins in the pore space.

A diffusion model for pulsed neutron logging

Based on Polyachenko’s formulation, the complete solution of the boundary value problem for pulsed neutron logging is derived for a centralized tool model. The solution is more general than Polyachenko et al.’s earlier solution in that the unrealistic assumption of zero thermal neutron lifetime in the fluid‐filled borehole is eliminated and that, in addition to the formation signal, the borehole signal is also considered. For formations of weak absorption, the theoretical solution predicts features in the gamma‐ray die‐away curve similar to those observed in real logging cases. For formations of strong absorption, the asymptotic behavior of the gamma‐ray decay curve is influenced by the borehole size. This theoretical solution, even though not directly applicable to the complex real logging environment, offers us a means for understanding qualitatively the interrelationship among all thermal neutron parameters of the borehole and the formation.

Magnetic field nonuniformities and NMR of protons diffusing in a porous medium

Magnetic field inhomogeneity can arise either because of an externally applied field gradient or because of spatial variations in magnetic susceptibility. The latter are most important when the solid matrix includes paramagnetic substances and when the uniform applied field, and, consequently, also the Larmor precession frequency are very large. Both types of field inhomogeneity add extra phase shifts to the precessing spins. These phase shifts vary with time and position in a complex and random fashion as a result of the diffusive motion of the spins. We have studied these effects by performing detailed calculations for the case of a fluid filled porous medium with a periodic microstructure. Special attention was devoted to the question of whether the statistical distribution of the phase shifts encountered in a Hahn spin echo experiment or in a Carr-Purcell-Meiboom-Gill (CPMG) spin-echo train can be approximated as a Gaussian. The mean square phase shift is measured in such experiments as an enhanced relaxation rate of the precessing transverse magnetization. We determine this mean square phase shift for periodic composites from the diffusion eigenstates, which were calculated using a previously developed Fourier expansion method. The enhanced relaxation rate depends on the echo spacing time tau in a way that can be correlated with important length scales of the porous microstructure. Those correlations can be extended also to disordered microstructures, like the ones that are found in natural rocks. We compare these theoretically predicted correlations with CPMG measurements performed on protons in laboratory samples of brine saturated sandstone.

NMR T2 Inversion along the Depth Dimension

The spurious signals in NMR logging often cause neighboring depth intervals of the same rock type to have dissimilar T2 distributions and oscillatory porosity responses. We found that we can use the 2D NMR concept to jointly invert T2 distributions of sequential depth records using the depth as the extra dimension and basis functions as the regularizing tool for smoothing, thereby constraining the solutions to have a reasonable behavior. Our results show fluctuation of both porosity and shape of T2 distribution is reduced. Such practice increases the feasibility of using T2 distribution as a rock type indicator.

Self-diffusion in periodic porous media: a comparison of numerical simulation and eigenvalue methods.

Random walk computer simulations are an important tool in understanding magnetic resonance measurements in porous media. In this paper we focus on the description of pulsed field gradient spin echo (PGSE) experiments that measure the probability, P(R,t), that a diffusing water molecule will travel a distance R in a time t. Because PGSE simulations are often limited by statistical considerations, we will see that valuable insight can be gained by working with simple periodic geometries and comparing simulation data to the results of exact eigenvalue expansions. In this connection, our attention will be focused on (1) the wavevector, k, and time dependent magnetization, M(k, t); and (2) the normalized probability, Ps(delta R, t), that a diffusing particle will return to within delta R of the origin after time t.